The following two sections describe the methodology for determining effective temperature and luminosity using computer procedures written in IDL. All IDL codes are included in the appendix of this dissertation.
Determining the effective temperatures (Teff)2 of M and L dwarfs has traditionally been
difficult due to the complex nature of radiative transfer in cool stellar atmospheres. The task is particularly challenging in the L dwarf regime, where inter-phase chemistry between solid grains and the same substances in the gas phase becomes relevant. Significant progress
has occurred recently with the publication of the BT-Settl family of model atmospheres
(Allard et al. 2012, 2013). The BT-Settl models are the first to include a comprehensive
cloud model based on non-equilibrium chemistry between grains and the gas phase and the
2Theeffective temperature (T
eff)of a surface isdefinedas the temperature at which a perfect blackbody
would emit the same flux (energy per time per area) as the surface in question according to the Stephan- Boltzmann law: F =σSBT4. This quantity often differs from the stellar atmosphere’s actual temperature,
rate of gravitational settling of solid grains. They have also been computed using the latest revised solar metallicities (Caffau et al. 2011). The authors (e.g., Allard et al. 2012) have
demonstrated unprecedented agreement between observed M and L spectra and theBT-Settl
model atmospheres.
We determined Teff for each object in our sample by comparing observed photometric
colors to synthetic colors derived from the BT-Settl model grid using custom made IDL
procedures 3. Our procedure exploits the fact that synthetic colors can be computed from
synthetic spectra and those colors can then be directly compared to observed colors. How well the synthetic colors match the observed colors is then a measure of how well the input
properties of a given synthetic spectrum (Teff, log g, and [M/H]) match the real properties
of the object in question. The best matching Teff can then be found by interpolating Teff
as a function of the residuals of the color comparison (observed color − synthetic color) to
the point of zero residual. The technique can be applied independently to each available
photometric color, and the standard deviation of the resulting ensemble of Teff values is the
measure of the uncertainty inTeff.
In our implementation of this technique, we combined our VRI photometry (Bessel sys-
tem) with 2MASS JHKs (Skrutskie et al. 2006) and WISE W1, W2, and W3 photometry
(Wright et al. 2010) to derive a total of 36 different colors for each object covering the spec-
tral range from ∼0.4µm to ∼16.7µm4. We then calculated the same 36 colors for each
3A thorough review of photometric quantities, terminology, and procedures for deriving synthetic colors
is given in the appendix of Bessell & Murphy (2012).
4We did not use the WISE W4band centered at∼22µm because it produces mostly null detections and
spectrum in the BT-Settl model grid using the photometric properties for each band listed in Table 5.1.
Table 5.1: Photometric Properties of Individual Bands
Band Blue LimitaRed LimitaEffective Isophotalλ Mag. Zero Point Reference
µm µm µm photon s−1cm−2
V 0.485 0.635 0.545 1.0146×1011 Bessell & Murphy (2012) R 0.554 0.806 0.643 7.1558×1010 Bessell & Murphy (2012) I 0.710 0.898 0.794 4.7172×1010 Bessell & Murphy (2012) J 1.102 1.352 1.235 1.9548×1010 Cohen et al. (2003) H 1.494 1.804 1.662 9.4186×109 Cohen et al. (2003) Ks 1.977 2.327 2.159 4.6692×109 Cohen et al. (2003) W1 2.792 3.823 3.353 1.4000×109 Jarrett et al. (2011) W2 4.037 5.270 4.603 5.6557×108 Jarrett et al. (2011) W3 7.540 16.749 11.560 3.8273×107 Jarrett et al. (2011)
For each color, we then tabulated the residuals of (observed color −synthetic color) as a
function of the synthetic spectrum’s temperature. The residuals are negative if the synthetic spectrum’s temperature is too cold, approach zero for spectra with the right temperature, and are positive for models hotter than the science object. For each color, we then interpolated the residuals as a function of temperature to the point of zero residual. The temperature value of this point was taken as the object’s effective temperature according to the color in question. We then repeated the procedure for all 36 color combinations, thus providing
36 independent determinations of Teff. The adopted Teff for each object is the mean of the
Teff values from each color. The uncertainty in Teff is the standard deviation of the values
used to compute the mean. After performing this procedure we noted that the majority of
colors producedTeff results that converged in a Gaussian fashion about a central value, while
other colors produced outliers that were a few hundred Kelvin away from the Gaussian peak.
Further inspection showed that colors for which the bluest band was an optical band (VRI) were producing the convergent results while colors in which both bands were infrared bands tended to produce erratic values with no apparent systemic trend. We therefore performed
the calculations a second time using only the colors that had the VRI bands as the bluest
band and excluding I −J, which also did not converge well, for a total of twenty colors.
Occasionally, a color combination still produced an outlier at Teff >> 2σ from the adopted
value. These outliers were excluded as well; however, the majority of objects had their effective temperatures computed using all twenty colors. The fact that none of the colors composed of infrared bands alone had good convergence emphasizes the need to include optical photometry when studying VLM stars and brown dwarfs. Most likely, the reason for the non-convergence of colors that do not involve an optical band is due to the smaller effect that temperature has on the relative flux between two bands when both bands are close to
the SED’s peak flux. Examination of color-magnitude diagrams with the I −J color also
showed a degenerate sequence where the same colors corresponds to a wide range of absolute magnitudes.
Figure 5.2 shows the graphical output of the procedure for determiningTeff for the case of
DENIS J0751-2530 (L2.5, ID #21). The families of vertically oriented small dots represent the residuals for the color matches between the synthetic spectra and the actual observed
colors. Each vertical grouping of small dots corresponds to a synthetic spectrum with Teff
given in the horizontal axis and contains one dot for each color comparison. As tempera- ture approaches the actual effective temperature of DENIS J0751-2530 the residuals become
smaller and less dispersed. If the models were a perfect representation of the SED of DENIS J0751-2530 all colors would converge to a single point when interpolated to zero residual. Because the models and the procedure are not perfect, each color interpolates to a different temperature at the point of zero color residual. The mean and the standard deviation of the
Teff spread from interpolating the several colors to the point of zero residual is taken as the
effective temperature and its uncertainty. The short vertical lines represent the mean value
before (dashed lines) and after (dot-dashed lines) the exclusion of R−J. The long dashed
lines represent the 1σ uncertainty.
The model grid we used was a 3-dimensional grid with aTeff range from 1300K to 4500K
in steps of 100K, logg range from 3.0 to 5.5 in steps of 0.5 dex, and metallicity, [M/H], range
of−2.0 to 0.5 in steps of 0.5 dex. The procedure was repeated for each different combination
of logg and [M/H]. The final adoptedTeff was the one from the combination of gravity and
metallicity that yielded the lowest Teff dispersion amongst the colors. As expected for VLM
stars and brown dwarfs in the solar neighborhood, the vast majority of objects had their best
fit effective temperatures at log g = 5.0 and [M/H] = 0.0. The color−Teff interpolations
often did not converge for grid points where log g or [M/H] was more than 1.0 dex away
Figure 5.2 Effective temperature calculation for the L2.5 dwarf DENIS J0751-2530. The small dots represent the residuals of the color comparisons based on synthetic spectra at different temperatures. Interpolation to the points of zero residual yield the effective temperature. The long vertical lines denote the 1σdispersion before and after the exclusion of outliers. See text for discussion.